The DT sheaf on shifted symplectic schemes and deformation quantization

Abstract: A complex symplectic manifold has a canonical deformation quantization and a Lagrangian submanifold (equipped with an orientation data) has a canonical deformation quantization module supported on it. Given a pair of Lagrangians, the Hom complex of the corresponding deformation quantization modules is a perverse sheaf on the Lagrangian intersection. I will compare it to the DT sheaf. More generally, I will explain a (conjectural) relationship between the DT sheaf on an oriented (-1)-shifted symplectic scheme and its Batalin--Vilkovisky quantization. I will explain what the implications of these results are for character stacks of closed oriented 3-manifolds. This is a report on work in progress joint with Sam Gunningham.